A note on one-sided interval edge colorings of bipartite graphs

Carl Johan Casselgren

doi:10.1016/j.disc.2021.112690

Abstract

For a bipartite graph G with parts X and Y, an X-interval coloring is a proper edge coloring of G by integers such that the colors on the edges incident to any vertex in X form an interval. Denote by χint′(G,X) the minimum k such that G has an X-interval coloring with k colors. Casselgren and Toft (2016) [12] asked whether there is a polynomial P(Δ) such that if G has maximum degree at most Δ, then χint′(G,X)≤P(Δ). In this short note, we answer this question in the affirmative; in fact, we prove that a cubic polynomial suffices. We also deduce some improved upper bounds on χint′(G,X) for bipartite graphs with small maximum degree.

Highlights

An interval coloring of a graph is a proper edge coloring by integers such that the colors on the edges incident to any vertex form an interval of integers; this notion was introduced by Asratian and Kamalian [6], motivated by the problem of finding compact school timetables, that is, timetables such that the lectures of each teacher and each class are scheduled at consecutive periods
Asratian proved that if a bipartite graph G with parts X and Y satisfies dG (x) ≥ dG ( y) for all edges xy ∈ E(G), where x ∈ X and y ∈ Y, and where dG (x) denotes the degree of x in G, G has an X -interval coloring such that each vertex x ∈ X receives colors 1, . . . , dG (x) on its incident edges
We prove that Problem 1.1 has a positive answer for every bipartite graph; we show that a cubic polynomial suffices

Summary

Introduction

An interval coloring of a graph is a proper edge coloring by integers such that the colors on the edges incident to any vertex form an interval of integers; this notion was introduced by Asratian and Kamalian [6] (available in English as [7]), motivated by the problem of finding compact school timetables, that is, timetables such that the lectures of each teacher and each class are scheduled at consecutive periods. In general, the problem of computing χint (G, X) is N P -hard; this follows from the fact that determining whether a given (3, 6)-biregular graph has an interval 6-coloring is N P -complete [2].

Results

Conclusion